Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}3x-6y &= -9 \\ -x+4y &= 8\end{align*}$
Answer: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $2$ and the bottom equation by $3$ $\begin{align*}6x-12y &= -18\\ -3x+12y &= 24\end{align*}$ Add the top and bottom equations. $3x = 6$ Divide both sides by $3$ and reduce as necessary. $x = 2$ Substitute $2$ for $x$ in the top equation. $3( 2)-6y = -9$ $6-6y = -9$ $-6y = -15$ $y = \dfrac{5}{2}$ The solution is $\enspace x = 2, \enspace y = \dfrac{5}{2}$.